Some Nonlinear Vibration and Response Problems of Cylindrical Panels and Shells

SOME NOELINEAR VBWATION AND RESPOPEE PROaLEKS

OF

CUEXNDRTGAE PAXgEU

AND SHELLS

ln PaePaO F d f a l m e ~ t of the Wequlrsments For the Degsae of

Large amplitude vibrations and forced resp~nsea of curved panels and ehells are studied b y the applisaasn of $he sh~lBos;r~ shsfbl equation. The GoBarldn proeedura 88 used t o reduce the n o d i n e s ~ par$;iaJ differentid aquatione t o o~&nary nod.inear eqnatf ens, Mp/~a~k~ed diflorenc e s are found betx~r~eea the behavior of cillr~ved i3weB~ and eylindzisd ehsPle. XelatEaono f o z the dependence

of $he

free vibration period

on

sn3pliguda ara given. t a o i m ~ d ~ ? ! a a d y a d s of Lhe cy1indricaW ~ i ~ e l l pr8bIdzm ?,g included.

The G U B V ~ ~ pme1 is fsuad $0 szAibl9 a backling ghenone-

saon POP &be ~ f m x p i e B%reathfn,g n-ao;ldeati, Shock response x~at"a10de are used t o predict dyna-mic buckling of tlaa curved panel a1;'kid

the pradictions are variUsC by n u m e t i e d integration.

T h e c y 1 i ~ d r f c d ohell vibratPoazs m d rBspoaees are found t o be g o v ~ r a a d by m f f h g 4 s aquacon and cert&n of the wall* kno~wn 12roperGee of *D&f$inggs aquati021 are applied t o the c yhiiadricd shell dparfif c o.

The two ~ ~ o d o a a l y e i s of &he cyUndricd shell lie shown ts a*bbit w e & c * c ~ a p l % n g , allov~ing the separate sxcitatlsn of &he c o ~ p l e d miode s

.

Intxoducstion

The Gurved Panel 2 r ~ b 1 e r i i The Response of a G U P V ~ ~ Pane1

A

Second Dehi.elop:~~ent of %he Curved Panel P r o blexn

The Gylindxical Shell ProbPa12~ kP

ah ~ v o :.;ode C y a n d s r :"analysf s

Gsnckasisns and %m::arhcal 2 e ~ u l t a References

i 2 ~ p p endlx f

AraApl%tude of vibration

3*~*~ipliO%zde of stress function; or aaipliitude

of ~Exx3nd n,od@ af viaration

dending stiffa@ s 5

Youggg e ~ i ~ ~ d u l u s

E n e r g y function %^or frss v i t ~ r a t i s n

'StresE3 T)uneelo~l~

Energy cona&anks

Shell or panel Bsngth

-? ending A:.o =iicnt re aultant s

Stress resultants

-,

i+~-\agni&uda of a step Pangstion in

R;

g19r 4

ratio of frequencies

-v

2 re$ vi ;rratioas. periods

Two mode tota3 phase paraxzete2~

L'%sP@G$ 3a;beao

Shock response ratio

Radius csf shell or panel

A parasetear: of $-Ifllts acjuatfoa

.Istability dateria,fnan$

C t r e s o function coeliicients far the t"~%~o clods analysis

Shell. or: pane1 G ~ i e k n e s s Sh!~ass yer u ~ 3 t area

e r 4 ~ g i a B :*-iods QUY:I$C~

C it c u t ~ f e r e n t i a l rzcsde nu,::^ g2er

Cartesian coordina%es

Chock ragpesnsa axxiplitude pararzlstar

Tax0 xm0de L o r ~ i a g i ~ n c t i o r ~ ~ a ~ 1 p p 1 i t t ~ d a ~

Psr;.sli:etes~ of the LWO ~~>.ad.e analysis

D e l t a function ai-r-plitudo cosffi'jiciaat Nondiazeneiesnsl frequency

Tvm mode analysis par=-~etsrs oi phase relation a; and c o a p l n g

\

-

-

&:ode n u r ~ ~ h e r s for the two 2usde analy~ilz., Poiassn'e yatio

m

A WCD YX 'ode phe s e relation para~-j eter a

N

s n l n e a s i t y parazx~eta r %or s cylinder 2 i o t i r - ~ i e ~

The atady of c y l b d r f c d aheU vibrations dates from 4884,

the eacond eation af Raylefghb fmoas

(a),

where ce&afa dicplacsms& modes are assumed in order to compute the as~ocfat~d frequencies of vlbra%fon by the opplica%ica

of

the

Lam

g r a g a equa$ions. apprwaeb Boo developed b t s %he widely used RsyPefgB-Rita ma%od of more recant li&@riitturs. A ~ e v i e w ~

of

the devslopmenta from the tfms of RayIeigh mtif 3957 is given in refsrenca 2 ,

To

that time R e i l s a h a r R ~ papa~33~ ( 3 ) was the prkncf- p1 shall~sv s21d.l study of the nnodlnaar v%"ozatioa~

of

~ y t h d r i e d \ shells (pmePaj, 3n 1998 A. S . VsL'mir (4) u s ~ d ths ohdlow @hell equation8 to study the stability of cylindric& shells (panels) v+i%k r a p f a y applied m b l loadfag, and in 1959 V . %. Agamisov zmd

A. S. Vol'mir 45) again ass9 the shallow ahaU esqmtiona to ~ 4 u d y both a x f

al

and hydro etatic load& which had beem applied d p a m f

-

coP3by, 3.a $964 Qug e paper ( 6 ) appeared with a d i e c u ~ s i s n

of

$he influencs of large mpIi%ud@ e on e y l h % d ~ I ~ a l she11 vibr&tioas again utilisiag the s h & $ ~ w obell eq&iiationa.

Even with these several papers uoiag the ohailow shell approach to study the large ampllgude ~bsa%ion or Pasponee prob- lems oB cylhdric01 @bell@ a d pmels e there v~ti?re a t i l l a n u m b r of qusotions &at ~ a m d n e d to be a s w e r e d

by

the application s f $be shallov~ ~hslkl equa%ions. Xt is t h e purpose oi tbf a thesis $0 80

-2-

abaPlo*~.r ahelf equaaons a m asad to sktdy vibration m d re- spomBe proble~na f i r a t f s ~ a curved pmeh, then for a cylindrical shell. A two -&node = d y a % s sf the cyUadrlcal s11slP l a 2 r a ~ e n t e d

I2.

CCF%ag$er VEI. Ia k l ~ i g connectboa% it s ~ ~ w a i d be noted that the L-JJO _, 1"1:mdc peoblai:-A l o r &he cy'$%HndrPcal shell 123 ~ i x ~ i p l e r t$ai the two

mode probSen2 %OE a curved plate.

The final chapter i g devoted Bs nuw~efi~iacal rasuB&a. Theee

- 3 -

G N A P T E R II

TEE

CURVED

PANEL

PROBLEBA

2 - 1

Ths psecsnt saalyeis will @tart f r o z ~ *&~9 V O D K a r m m large deflection plate equagion e ~ e n d e d t o include an initial cur- vatuse. The ccoordinats system and nomancl6turs are repre

-

sentad i a figure 4.

A

panel (plate) of length

L

in the x-dir~ccti~n~

width (7ia/n

)

in the y-direction m d thickness h in the .-direction. initially lies in the xy-plans. The xy-eosrd&ate syatexn is

located wi& respect to ths panel sjrueh that the region of tAe xy- nd L ycLa O O - 6 L , The p1me covered by the p n e i Ha

-

_{z* J}

plate i~ given a stmall positive v e e i e a i displacement QpooitGte di@plocsmeat is in the upward or %-direction) to form a a h d l o w cyliad~icafk ourface

-4-

Eere the operator

v4

is defined by

while the ~ t g e s g function

-

is related to the strees resdtmts

/d

.

N

y and

Nx

by the relations

Whan

s

cylindrical surlaca u d e r mgorrfi axial or circkw fsoential ~ t r e a s $8 to be etudied it is cowentent to separate the

constaa~t i n i t t a ~~Dtress resultants N r i ~ u a d N ~ i trcm

Nx

and

E\ly

by defhxlng a new s t r e e a haaction

When squagions 2.4 throwh 2.6 are used togather with the sirfipli- fded notation

Some d p m f c p ~ ~ b l b e m ~ of very t h h w d l e d cylBndriea1 panela will ba considered. 1% io aoaumsd in these problems that

the deformations ( w ia the s-direction, u in a s x-direc%ion, v in the y -direction), the flexuaeal deformation, w

,

g r s d s m b t e s . In reference 3 it hoe been shorn th& for the lowera (fiexural) mods@ of f~esf vibration oi thin cylindrical shelf s , wban n is nag toea smd11, say n

>

3, then ths mswiimw hsrtia1 farces due to accelerations tgngent t o ths surface of the shell ( i . e.

.

;GI and

' *

-

V ) may be neglected in c o m p a r i ~ o n t o the maxim- trw=m-

verse iner%ial force (

w

% 1.

Thia ability to neglect the

u

% pnd

$ 2

airriplifies the analytical problem tremendously. Wama ws shall a@smSe, explicitly that the only ins~ial force; *ich i o to be conaidered i~ that due t o the tran~vsrse accel- eration

2

Under this assumption the load n o r m P tc gunfacs i o considered to consist of

where is a uniform constant pressure and

PC%,%

t ) i a a space a d time d8,wnd~nt preseurs~?. %Then equation 2 . 9 69 LZA-

tkaoduced into equation 2.8, the resdt is

.

.

D

v 4 S

'

( R - 2

h ~ q i ) f

Pc*,y;t)

1 4 I

1Sf% the

curved p m e l ~ ~tudied h a r e w s r s s part aoPs pressurized cylinder d t h o u t other initial streeBes, the ~ e l a t i s n

N

i =

F?

d would hold and the t e r m

(

P,

-

Nfi

)

would be z e r b . Other- w i s e , the squatloas of motion are equations 2,10.

2 . 2

W e c t a n g d a ~ pmsl~ subjectsd $0 ths ao *called ''greely

@ugpor%ed'%mdasy condigion8 will be c o n ~ i d e r e d . These con- a t i o n s are

s

=

S,,

= o

O N x=o,i-

,

F =

Fxx

=

o

Tne eowditiosr,

Fgx

= 0 on a b w d a r y x = coast& reegetfres that

N y

be eero on that boundary. The additional coedition

F

= O on the b w d a v x

=

conatmt requBres that Fv y = 0 or

f\lx =

0 on that bomdary. Since the stress-strain rsla%ioae way be written

G

h

Exv =

N X Y

,

-7

-

stress resultants

N

and

N

and the aormal strains

a?d

vanish sn the h w d a r f e s while no r e q u i a e m n : ~ ore imposed on the shear o%res&aes and tbs shear s t t a b s by the bowdary can- dfgiono.

h

a. @ir&i%ar mame$ the $ o u D & ; ~ P ~ ~ o n d i t i o a S = 0 OD x

.

const- r s q u i r e ~ that

Syy

= 0 on x

=

constant. The relation between the mO338nO res~jhtmts a d the dispbcerfian$, S , axe

Thue the bomdary conditions an S psovida tan additional csndition that

M

x and

My

be? e@FQ on the b o w d ~ i ~ i e s bit MXY is not pzs- 8 ~ ~ 3 i b e d the Lw%~da~iea.

2 . 3

A p p r o p r i a % ~ E ~ O ~ B B for the dfbpbtaeement fmctian S and the

stress fmction

F

must be selected in o r d e ~ to apply the Gslsrkfn m t h o d .

These

m d e s must satisfy tha prescribed b o w d r y can- dieioas, equations 2 . 4 4 . A set of modes \ v h i ~ h ~ a t i s f y these baud-

ary conditions (as *seiiell as b a b g a 881%1&ti&3n to the lheariaed equation o&ahsd when the aanlinsas t e r m s are drsprsd from equations

2.18) are

-8-

The choice of thee@ w d e a c o z ~ i e s wilh it the in2glbcBt assumption t b t a$sy nanltnear%ty in the problem wQ.1 L a w n e e only the a a t a p ~ of the time depeden% mplkiftades A(t) and

Be)

m d will In no way

affect t h e space distributions of stresses azod &splajacsment~.

As

m approximati~n* this a s a m p t i o n can somstbee be justified en on sxpssimeatal basfa.

The Qderkin p~ocedssre rsquires that the m d s e (equation 2. 914) be subetiguted los S and

F

in the eqaationa 0% a2otHon, 2.64;

that each of ffhsas equations be %vi%reightsd ~ ~ 5 t h the a p p r ~ p ~ f a t e modal function (in lBia case coo(%) stu (?)

1

and that the reaultmt

rrd

-"a

s y 4

z m

e+ation@ be in%bgra%@d QV@P *6 don~ab? 0 O )( XC

,

_{A n}

E v d w t i o n of the integrals

0s

equations 2 . 2 5 a d 2,16 give@

(1"

2 8~

-

L B ~

COS'

( 2 )

S I (r) ~ A X J y

-

(-

),

0 -&V a m

/ / ~ c o s 3 ( ~ ) s , ~ 3 ( ~ ) d x d y =

(%)($y)

-10- Then 2. iS becomes

and equation 2.46 Beeamat?s

lf

equatfoa 2.18 $8 used t o elfmfnata

B

from equation 2.49 and $3

@

the f8oUovxfng notations are used

2

the equation

i~ obtained for tho nonclimeneional amplitude coefficied

{

.

2.5

h

reference 3 R~isaaraier developsd, by an eatfrelp dugex- eat m s & d , aa squation identicall with equation 2.21% ~ x e e p t for the la@% tef~n3. R e f ssner consider 8 an initially pr ea s a ~ l a e d panel

which bs +eprsssnted by t h e shallaw ebell eguaLfoas, 2.8.

Hence the, condition

N q i

=

a

is a s s u e d BO thal _the last term

h

equation 2 . 2 1 v m k ~ h s a , m i s sgreems& between the reedts obtahed by G a l e r k i ~ ' 8 me&@d and Weiasner" s e t h o d provides

@oms esnfiijidgsnca In equ&ion 2 . 2 4 ,

R ~ f a aner, appiyhg the L i n d ~ t e d t - h B i n g perturbation L s e b i q u s

,

s'&shs the ]%oU~wHslng e x p r ~ ~ ~ f o n relaeing the free vi- br@ion f requeac y and mpllitude

where i e the w p l i t u c i s at 5 0 .

,

.

.

.Oh@ r s m a r b b l e fact that ahell does not gpend equal t h e ka%ervals deflected outwards and deflected inwards. Rather, more U n half sP the eyc1e 18 spent during the7 h w a r d defiecfion.

m p l i t u d s

.

2 . 6

h.

Chaptar

UP 6 e

a d y e i s sf a con3plete cylbdrical shell 1s d i s c u s ~ e d in detail, ~ U it I i s ~ u f f i e l i e d to make o d y a. b ~ f e f comment here, b order to @&end t h e appllic&ion or% the @ h a l o w

shall squaticano ts thai3 csmplete c y l h d s r , it is necessary to chmge $he limits of &e G a s r k i n btsgragion t s hclude at Isas$ cne f u l l wave, i s @ . . - L L x L L

;

:

Y

5

3

.

Can~equently the reoalts cgsn~fdered fa this chapter are mt ddrsctly applicable to a e complete c ykbdrical shell. Z t will turn sut Ohat f a r the corn- plieee shells $he quadratic gaonlhesrity vmiohes with It goao

This squation may

be

rewritten

as

where

Y =

4 .

(d,

0

)

.

where

( = o ,

If

and are complex, then

$

i s s .table center end the onLy singularity g l k n e .

If

and are real, i . e . , %t

then

(

_and

4

are stable centere and is a saddle p o h % [figure 21, Ths physical Hnte~pr~tatig~n of the i n ~ k a n c a e in a~hich sqwtion 2.26 holds l o &bat @table vibration@ may e x i s t abs&:

4 ) the w3deaected e q u i l i h r i u point

I(;

with a limit on the maximum amplitude.

1

(

1

-

1

,

1

.

at which

suck.%

o vibratii~n may OCCUI;

2) the 98buckled'' equilibrium p i n t with a limit on Ule a i ~ ~ i m ~ r n amplitude.

1

$ ( y 1

1

-

1

1

a& ~vhich such s vibration may ~ ~ ( t u r k " ;

3) tbe saddle point

,

end rsarcomwssbg Lhs two equilibrium poLnls

4

and

4

a d with a limit on the m l a r i m u ampliguds

(

d n ) -

#z

[

a&

wMch such a vi-

bragion mag QCCUZ e

t e r m

DL(%)'

+

(g)L

1

*

will have a s m d l ecn&+ibaicn compared

4 2 -Z

t o the zqembrane term

(9)

( 2 )

[

(?)=

+

(2)

1

,

unlees

(4)

>9

(f)

.

P

,

And indeed, Lhe initial pressure c a a be made ~s large that i%hs &P;llaquality 2 - 2 8 i o never true a d than no possibility oaf a buck- led @Bats exist@. In t b ~ t event the vibration 5s a b u t ths single center = 0 and is very close l o the linear sysLem in its bs-

a,

sa tbs other Band, this pressinure i s nsgaeege Qextsrnd p r a ~ a u r e ) it may approach the v d u s

which ~ ~ o r f e a p n d s to buccMing of this cylindric& panel w d e a ez-

reowndbg to

E

<

0 is not examined hare.

Integration of @quatien 2.24

n n

NOW

I

the defhttion of Y is restrfctsd to the s p e c i d case,

the freqasacy d, is replaced by it8 asoocia%sd period

-14-

&en the resdt

of

@@paragion sf variables m d htagratboa is

SLme the system described by equation 2 * 2 3 is C O E ~ ~ S B W ~ - give, the h d f p e ~ f o d of the a y ~ t e m is represented by sqaatfon

2.31

il

$

and

f2

are the extremes of m p l 9 u d a of the syBtem rspressntad

by

sqaation 2.23 a pa&%edar v d ~ e of K

h d so the ratio of the period of one compiete cycle of the non- Ifnsar oacflktfow &e the pe.rriaod of the equivalent l k ~ a r system i a

Equation 2 . 38 is as alliptic ii%teg~s1 a d its evdaatisa is discassed in AppeaBh

X

a ~ d ghe results

m e

p ~ e @ e & e d in GhapQdr V. Equa- tion 2 . 3 2 c a n also be used ic describe the p b s e plane ( $v

-

#-

plarme) trajectories and, with set ideneicdly ~ o m , to de- eerfbe &ha ~ m x f m m di6splaikeements far

a

given energy 1 ~ ~ 1 . b fact, if ki?qu%ion 2 , 3 2 5s written in form

The general cha~actesr

d

the phase p1-e trajectories, determbed from squation 2 , 3 % , is con%rsllad by %Be two p a s m e - Beas

E

d

K

which r e p r e s ~ n t fie dags~bz: of ~ ~ n i i a @ a r i f y m d the energy level, respectively.

E

@@ation 2.27 holds, %ere

ere

two stab&& c e ~ t e r s . The curve8 which e3$i~cl0sb efther of theae

*

c e ~ t a r a are Ein tarn e a ~ e l o ~ e d by the osparatrk

.

If

the molfon i o at &a energy level less t h n that of the s e p r a t ~ i x ~ then hf9lal con&%ion@ *11 debermine the esnter with ~vhie3B the motion f a a@- eoceSa%ed.

If

a$:@ snsrgy level f @ greater than i%h& for the s e w r a -

v ~ b s r s the upper sign c o ~ r a a p o d a go the upper h . d plme.

8

the v a u e

of

E i s Ism tbm 8 / 9 , the slope of 4&sss t r a j ~ e t o r f e l is

zero only at = O

.

But the madmuan inward defieclion. {in mex, has a greager absolute value %ban does /out m w . Xicw t h e tieme of t r m ~ ~ i t f r ~ m ni{ mtu; tc

/=

0 and the time

of

t r a ~ s i t frorn

+=

0 to

d

OIL% max are determined by

@ The saparatrix i s h t e g r d curve passing through the sadde

4

in tbc Wger h a pIkae. For ~ 1 e h v d u e of Y fa Bhe upper ha&f plane, there c c r a e a p n d s one value

cl

4

L1 each of the inward as~d outward denection states. Since the dimtwce

I

4

in max

1

eovsred by Che integral

t

,u is grearer than the d i ~ t a n c e

I

'/

oul max

\

Dhen

t,,

>

t

,

.

A similaa argument may b made far other trajec- $oris@ w h m & ) 8 / 9 . Thio result ha@ d o o been verified by direct n m e r i e d iz-Ei;ag~&in af &e egutfotl.9~ of m ~ t f o a . T h s types

of

-19-

GPIAPTZW

HI

TI-3E R E S m N S E OF

A

C U R V E D

PAYEL

A

few

d

the :,my method1 of mdyzL: t h e F B B ~ O ~ B ~ of

a n e n l l n ~ a ~ system w$kl now be applied. Tbs dgferen%fal equ,atf~n

malyssd is

Sfnee the discuooion of the Best eection has ds&% v ~ f & the p i a s @ p b s

,

a re@ponse problem which may bs b a a e d d t h the aid

of

pilase plane trajectories will be conaidered first.

1l

('1 is a ddsla f=c&fon

and

kf

the hitid coadigiana aaooeiated ~ 5 t h the problem are $(o)

=

0 , # T ~ o )

=

0

.

a new problem dhich is

e q u i ~ a 1 ~ t to

tab

may be bo~mulslited. I.n $be new problem Q (T ) = 0

far all vduse

ol

f

a d the abw hit'aaB, conditions are 4 ~ o . I = o

,

&

Co) =

3

.

The relation between d,

3

and a p r e a a u r s pulse

of

raa~imitpads T). i e

-209.

gpecfal probkem of ceenoiderabfie intarest i@ %a find tbs magnitude

of

for which the t ~ a j ~ ~ t a r y will e;acLo~e Ohe bucaed equLTPB- rium point

$!!,

( E 3

$

).

This value

of

munt cop reapend io the valocity intercept of the esparaat.6~. Thus the cri@ic& value of

pa

will ba d6termined from settinag

Ths equivdent statement i c

h

t e r m s afthe physical p & r a m e % e r ~

The eqlmatfon oL the sign in equation 3.6 emerges from

s consideration of the effect8 of d-pa'%in.,g,

k

t

the obsenee

OP

d w p - ing, the phase plme trasedsortao ass syrametrie with reogset to

-21-

the system os easily as woaalid

rn

ek%eea&

PXBSBUP@ p d s e . Thf s

48 no% phyefcaUy reagonablo. A mare ea~efa3: examhatlg;ln Se Ln

*

Q P ~ @ F . ~ x P B F ~ ~ ~ ? s B ~ & W Q Z ~ % b $ ~

vtateb

2%$; C.

2.

T . hat3 indicaed &at I%@ dhmphg c,& %he vfbrationa ~f thin eiia"~*&ar c y f h - drical shells ba sa%&f-sqa%dva1eat fo figgy oz o n @ - h m d r ~ ~ d ilcyelso to

b m g to ha@ m p f i i t u d ~ . D m p f n g af such emaU magnigude i~ B&*-

iafwtsriiy traaged as viiecou~ dmpbag. Zf a, visiicous d m p f n g

tsleem i@ f n t ~ ~ d a e e d hgo aquagion 3 . 2 3 ,

QBe

phase

p h s trajactories

ape na 1 0 n g e ~ l&s figure 2 but are 5l;qov~ l i b figkrlras 7 m d 8. F P O ~ ~

figure 7 it is elear sat for zs, trajacatory to enter tha bucMad ~ s g i o n , $ha @quivamt hi%%al qir@loef%y must

be

negative, that 1s h % v a r d + Hanca, $&,a prsoaure pulse muet bsis eeer-l. r ~ g u i r e s the

3 , 2

It $a worth n o t b g thst the phase plma tzajectories provide

a d~B;ermim$ion

QI

tha m a i m - amplitude response, of tihis @he81 fiiystem, 80

an

appi%ed impdsiv& pre~gaure L~adiag. The i a W i & v@locfty, corhaspondfag BSo

a

preeoure pul s 6 # idsatifiea a given p h a ~ e p P a e trajeetoq. The maxim- arnplJL$&adg? asso~fated wit32 that %r&j@cQov i s the maximum eb8tJakie deformation tha* the shell can experiene e

.

%B o&a.&;&cked. Csasldet the iteration ~cherne

h g ~ ~ & t e @qaa%$on 3 , 7 h t ~ equatfon 2,223 d g b t h e hitial cendi-

tions

f!

(0)

=

0

,

0 )

=

.

Acd further ssaurne

r b t

4

_{(0) =}

i,

is a eonstant. Then is determined by

t h s +o&s

are

&s earn@ as &a;ssa gowd bw equa%iarl 2- 25.

W%en

4

=

$

+

6

is intraduced into equation 2 . 2 3 , there follovrs

4)

f r

+ s : g ,

I

I + * &

8,

(I+-

t8,)]

= 0. ( 3 , 9 1

Tbs s o l a ion $0 ewa%isn 3.9 f 8

Mter apptieation of

%he

initial conditions to

,

.

the ewreosica2 obtained is

&,(TI =

40:.,

[ I - L ~ S P Y ~

+-p3

S I N ~ Y

.

(s.azj

-23-

has been matched to i t ~ e l f .

l

2

do,

is %&en ts be the b.acMed squilfbria% p e t t i a n (gar C 2 8/91,

then ths condition,

deleminee Lhe values of

d*,

and h vvhich are

an

acceptable sclu- %ion 0% e q a t i s n 2 . 2 3 copreepsndhg t o equation 3 . 4 2 . Fi"%gu~s 9

s b ~ v g s several curves repregenting ( ~ / G L ) ~

a

f m c $ i ~ n osf

lor @ever& valu.uee of E

.

% is cbsewed that as 6 ia-

creasaa

above 8/9 that

a

larger m d i a ~ g e r part of the c u w s in the r e g k n

-

3 C

d0,4

0 ccrreaponds to uaccep$able valuea for

)

a t

do)

4 - 3 these&Kilf always $ e m acceptabBe solagion of the ~ Q I E P ~ of e v a t % ~ a 33. 42. F O P f n s 4 m e ~ II BI & be-

comes unbundedlp large, the

value

of +(

*,

tends t o -4.5 and for

t

=

1. i e - 3 . 2"h~he region co

>

C ?

1

can have solatioris ~f %he farm given in equation 3 , i Z far V I ~ S ~ P ~ ~ I O B S a b a t the k ~ a e d eqablfbrim 4m9nt given by sqwtion 2.53,

to a step fmction with zero i n i t i d con&tisns irfiwsad,

%.

e ,

,

d

(O)=

-

-

₍₀₎

=

₀

,

_{Q(Y) =}

R

_{for ?'}

3

₀

,

_{where R i a a c o a s t a t .}

-

24-

Again %he i t ~ r a t i o n taeBnfqus is applied, s e s k L ~ 3

a

Ilneariasd ealution approximating the action

of

$he a o z ~ l i n e a ~ sy seem, Ths forn3

of

the iteration 18

vrh@r@

&,

is

a

c o n a t m t . The

Q,,,

m ~ s t satisfy

vgith

f

de&oed by equation 3 . 1 1 .

To

determine the range

of

vslidity of equation 3 - 4 9 , il i~ a ~ s u m s d

Phe

i)

( r ) is v e r y nearly fie cor- rect @ o l u t f o ~ , &6n the true oolutioa,

tT

becczxes

k,,,

+-

4%)

La-

? = ~ , p r

+

q c o s 2

I=

~ ~

o.

e3.231

the foram (ref. 7 )

From thfo I

a

b,

- ~ q

(b,,,

+b,.,)

+ ~ 7 n 3 , ( ~ m + , + b , . , ) - ~ ~ + ~ p ) ' ~ ~ = 0 ,

I I ( 3 . 2 6 )

[ a - ( ~ + ~ p r j

b,

-

I ? = (b,,, + b,.,)

+

.i:

(

bm+,+ b w - r ) = O ,

i s obtained. i ~ s o c i a % a d with these sqyzatia:~g,s there i s an ind&nPce 4s- eesmiinmt vvi%h $be ~ b a t ~ 8 1 seven by S ~ V ~ X I dets rminant, Tks condition %hat t h i a determinant v a ~ i s h (OF far that. matter, any finite cea~itrd

determinant taken from the idinite dstsr*minxat) ia an app~oxkx2aOe condition to d e t e ~ m i n a the values of md i26fi~:e the stabiB&y

of

- 2 4 -

then

4,

is n o t a good appp+o.xit:iation to the t r u e anlution of equa- tion 3. 15, It ls certainly that the vanf ahiag of is the imundary

bshwesa stable and unstable values of

&(,,

.

biaking u s e of this Pact the first order stability criterion for

hi,,

i s

Y,

=

=:[h

8,,,1.

-

*

iqherra $he i:necessar$p opsra~oas have !3eei;n carried i n $ ~ g . ~ a $ i ~ n

3. 27 '&kc conditions lor stability oi

d,,

and hence (he ?ii;.it!ng conditions ccder vrlzich

J(,,

can represent tho raspoxace oh the

-1

v Q,,, 8s defined by equalion 3, 1 7, -&qu;ll.Bf~ns 3. B '7 and 3 , 8 1 have been used $0 C O ~ S ~ T U C & a stability ~30~,1elgraph v d ~ f c h is

presented in f5g.u-e 13.

-27

-

A c o m p a ~ i s s a with equation 2.39 SBOVJB tba% tiw eEecest of a pressure stop fmunclicn i e to

add

a rsrm, linear in

#,

tc the elaerpy equation

p r w i o ~ @ l y ob*&in@d OF the free vibration, These energy 4 8 ~ ~ ~ 6 8

srs

@horn

figure 5 . A8 e>cpectsd, an. SermP preosuza step bilizea Oh@ cygfnde~ a d

a

s x t s r d

pressure

step destabilize^ &%a c3~liadsr. IEa addition a there &re &her PBS B 8 1 p ~ t e d F ~ B Q P ~ SI V2ith

m

e~ncesmd pressure step,

&he

sadae g s i ~ a moves closes to e h ~ origin,

#

=

O

.

and to a lowez energy level. The buckled egunbb- point move@ to a 1mer energy BmeB and a larger bvzsrd d i a - plaeeraent. Tha static a q u l l b r i a ~ pbt m v e s to a. higher enBrgy level and m b%sva~d displacement. hhem101 y x e d s ~ r e has precissly ths,s o p p s i t e sBect, Figure 3% showe how pl~ase pptam tzajsc- tsrieo are ~~3odBfied, Figura 14a depicts &a ~ a i n i m m snesgy bucaiz%g trsjactcry (ssporatrk] for a given system. Figure 41db

showe Bow this t r a j e e t o ~ would be ~h-ged by an. exterrd presgure step xvBfls figure Q l c abowa the M u e n c e of m infesrwl pressure step.

3 . 4

h

4nqpart~at m d in%ereating q ~ e a t i o r i %a ~ a i s e d by tbfe inQuence of

&he

etsp fmetian loading. 1% iis clear && the largex* the @&@FD&P ~ t e p f&"~cfion~ the l o v ~ e r be the

~nergy

level re- g ~ i ~ e d to C 8 U S 4 b u ~ u h g of $:he aheU, wail &@ st,@~ ~ ~ B B B U P B i d

e q u d to the otatic =mcMhg preosure i.n ~vh%cb e m s the @hell wW bucae;r M@lo& dPamic e f g e ~ t ~ . The qusatfoa that arises Pe "%is:

of magnitude %sea than the @fta$ic h a ; M b g preBsure;, wi11 buckle &he sbeflPP To mswar this question it i%e iislecesaary t o k n w the rnawimuna v d u e tBac

/

will attain

b r

any e x t e r n a l preeaure step and t s c o m F r e this value wi& the lacation of the @addle poht.

U

this m a i m ~ m reawnse i s greater thm t h e m p l i t a d e which cosreopoirad~ wi% the ssdde point, hcklilim i s gosefbls.

Application 069 the weark of F um d ~ B ~ f i o n (81 for the

@pons@ of 1% n o d i n e a ~ system to tba m a x i r a w r e ~ p ~ n ~ e of the equivalent &hear oyslem (t = 0 ) when each has been sabjectsd tto tbs oams Ioadbg, Thie ratio depends on the mture 6f the nodinaosity a# $he fe8L0rfng P O P C ~ in the s y s t e ~ ~ ccoasidered and on tha type of Loadkg to be applied to &see two s y o t e m ~ .

F O F

a step load, they have shown that the response ratio

%

has the foliowbg ;iom

-29-

and A i@ the maximum

' '

afilowable" D O ~ - ~ ~ F ~ S O O ~ O ~ & deflection

of the noaliasas system. me choice fag A is that deflection which is %he Isast that w a l &low bucHbg to occur, i. s .

,

the sad- dle psiat. The crftied loading ratis obtained ist h i s m m n e r fo

The maximma response of thia l h s a r system i s 1 2 P = whx 1 4 ~

\

1 a ~ ~

But recalling

$ha$

the m u f m u m nonliaear respowas to o ,step input of magnitude

r

is fa-d tc be

-30-

T h e three roots gdf this equation wf ll be ordered in the faahion

dc3,

dlal

go,

The magnitude of

(Io(,,

must correspond

It i % s not poaaible t8a&

an

b%:ernd ~ I P ~ B S U F I Isading wf10 gsser&%e a dynamic rsspoass great enough to buckle t h e ehall, G%us an ex- tern11 g r e @ @ u @ step may be taken as the phyafcdly importmt condition. The sh%ernd Loading situation eorreapondo Lo

r

4 0

Ef

equation 3.38 i@ Wrsduced h & ~ equation 3 . 3 7 af%eg dropping tihe a b s ~ l u t e va%1~@ eigns, &he bqwtion for value of

r

which will j u ~ t allow buc&Uing is

The roots of %hie aquatian are ordered as

cz,

I J;,, Ths vdua of

,

%&.I1 provide the correct m g n f t u d e

(and

s i g n 1 . f ~ ~ t h ~ pres-

8

sure step required to buckle the cylinder, provided only that 6

>

9

Figure 39 shows t h e varlaeion of f'(,

,

a8 a function of.

r

.

Z

Core muaQ be g&cen at t h i ~ p i s ~ t since

d L

depend8 on an t a i g i d

preaaure, The b a e m d presoure is not 8 part ~f the initial pre8-

(ST )

-

32-

GMAPTER 1V

A SECOND DDEVELBP&LZNT

OF

ThE CURVED P A N E L

FROB=&&

4.4

h r s f s ~ e a c e 3 Rsissncr a s s u e d a stress fmcbion that sat-

f sffed %the b u a d a s y condition@ bug w h f ~ I 2 did nab satigfy &@ earn-

wtfbiEity aquaefon exactly.

h

section 2 . 3 et seq. the s m s phpo~eduze %o fallowed, The squa%%oas of motion will bs o&ahsd uofng

an

a@-

@urn ad dt splac @ m e n mods shape md the C O ~ Z P P O B ~ ~ P ~ ~ ~ exoet st~438 8

fwction (which a a t i ~ f f s a the ccompatibility squation fa a mmner ~ % m f l a r 80 the v m ~ k of Ghuo

(6)).

@ t h e d f s p f a ~ e m ~ n t G[xlygt) B B

given in equation 2.44 f e used a d eu$o&iBat@d into the eompattbflity equatioa (&a f i r e & i%sf equation@ 2.8) the strseas fw~ckfon m a y be dets smiaed by integration of that equation, T h e Integration give o

This otseee fa~%i$tisn satisfies the eonlpatibili$y conditions exsc$ly but the b w d a r y coapllditfono on

F

a d 108 sscoad derivative@ are not satisfied exactly, but ase eatiefied only- ooa the avs~ags. &:one

the la@@, ~s gi?ssuI%s ~btafaed are eqeeted %0 be in9proved if squa-

tfon 4,4 is u ~ e d with the Gal/e~kin averaging technique 4 0 satisfy the s q u b l i b r f m e q w Q i ~ n (fih04 of e q u a $ I ~ n @

2,

$0) ~ d $ r the res$rfc

-

tioas

af

seetion 2 . 6 .

'When eqaagions 2.44 a ~ ~ d 4.1 are b t r o d ~ c s d info the first

(4.2)

-2

2 4 & . 2

.

L o s 2 ( T ) -5,"

-

3

(P)

( a )

[ K T +

($)'I

S I J ~ ) .

2 4

.

~ a ( ~ ) j

+ E ~ A '

(g)

C~S(?)

-

(2

j L S ( 2

st&

(TI

")-.I%

2. %7 a r e used slang wit& the hteg~alis 9

an

2 R %

( F X ) ~ , ~

(-.)cot&) dx

J r

=

( s )

and if the aotattons of equations 2.20 are uoed, ths noadimsnaiand aquatien obtained Irsm equation 4.4 Sa

then equatioa 4.6 may

be

writtan h t h e fora~i

4 . 2 Tbs Wluencce 0% Assect R a t i o

- 3 5 -

A? tends "k zero, 6 %and% $0 z e m D U ~

2

tends t o plus Irdiaity + - 4

-

1 s n d t i a s p ~ o d u c t % c ( R ) Z ( 1 9 ) tends to 3 ( 1 - v z ) ( r / 4 ) n (hh)and

Lor s:nall FR (CRc< I ) equation 4,Ei is

[.%.

no)

TEis,lu is the Daifing e q ~ ~ a t i o n vfft%l a hard spring and hencal: ~~d13e.,n the

%ion L, 23a =lay be replaced by equation 4. 10.

T o r tanding %o indinity C tends ta zero as ~ - 6 v d ~ i l e

a

X tez%ds $0 negative i-diniw &as /a { ~ e e figure $ 2 ) . The product

(2)

6

( a )

x ( @ )

t o m i s $0 zero 9 5 R-% for

2

diflerent fronl

But if i a only slightly d i f f s r ~ n t Lrsl:-~ the suckilia3g cond&isn

4'

%,T the product ( g ) E -% can becsn3.a fiaita and thr: equation of x2ga%ion reduces to

wkiiieh is Duf$ing8s equation for a a d % spring. binee eq~lation 4. 1 % requires a very restrictive s e t of conditions on and f i a it is

oB liriizited coeiulnes s. Hn S:LG~S$ instances it \-?ill be .mare a-pproprfate

to : %;e e q ~ ~ z t i o n 1-Z. P.

P"" "

1-ce co;-2?arf a s n of equ.ak;ioa

.-.

G v~5%h eq~atlasn Z , 2 31 s?n&iea$es

#I %,=$ a .-, the two \ ~ ; l % zive ~ i r z z i l a r results for &f? h : ~ the neigh-)azbood

61

sf 0.

9,

ki,o carry suttghc studies of s e c t l o a s 2 . '8 thrk-ougk 2. 183 f o r

-

36

-

a$: equation "L $.+ sections 2.7 thmagh 2* 10 may be regarded a.3 de- p i c t b g mmy 0 8 the f e a t ~ ~ @ ~

af

equatisa 4,8.

A

brief e x m h t i o n of the eP4ee9

oae"

/cR

on Oh@ s b g d a ~ f % b s s

of equation 4.8 Ivdth QC?) = 0 ) inthe #as@ 4 1 1 paint out how t h f e s q a t i s n d i s e r s from &at bte;i&sd hi sec$ions 2.7 &=ugh 2.10.

The phase

p l w e ~ h g u l a r f t i e a are the m d e of the equtfon

4

and are stable centers whQe i o a saddle pi&. If the o p p o i t e L ~ e q a a i t y

From these roogs is found tc be

a

sgablu center wliute and j(3 airs saddle poW8.

h

inimpohbad fea$.take oi' the large aowiect

-'/z

aagio problem i o &hat b t h 6 ( R - + ~ ) axxd

-

are small n ~ m - b r a so that 8 s tsnda to idfnigy amplitude of the staaa

oacillaticns about Ehe pohk diminish to

zero.

Thus

tho oacil- Pa%fone a very Bong ~ P B O W pmeB B~CQZ'II~B Lees 8fabPe as the

lea&& of the wne1 inereas@s. Figure 43 ahovcv's the variations in the 0me oO pbaae plme trajsctorie~ th& m a y be expected ts secozz2- mny cbaagbg aepsct ratios h.1 @quation 4.8.

%Be

k&omogsnsoao duferenaial equation wiCh homogeneous hltial c o n d i t % ~ a s considered in aecgion 3 . 3 caxx

be

treated sxactly

by

a

sLmp%a tranefo~~matie;an. 331s dsferantfd @qaa&ion than t ~ e c o ~ % e @ hsmogsnsous while the hitbal conditions bsco~me Womogensoua

.

b d It& bhe initid conditions ba

$ 0

=

=

0 .

NOW let

h

=

6 +

B(z1

where

6

i e a eora~tmt ~ U C I ~ tb&

-39-

M a i m = amplitudeo may

be

dsterz~ined from %he phase plane E r & j @ ~ t ~ p i @ ~ as a fwetirsn

of

&

.

A

given .%p& U B

of

W

will

Pix

a

value of

6

and

benee a given v a h s ot K, from

The period aasociiated with a;qua$ion 4.22 i s determined

5.1

It

hao h e n p b % e d out in ssctioa 2.6 the mamki$~ 432

which ths Gderkin av8~1gIng ~ P I Q C ~ S O f~ applied m&@aa a f.&adaw meat& B2ferd~ac&? in the *tare of the3 pby~irl=d sgsten? that ear- ~ s s p n d s t o t h ~ resdtbag squation. bdeed, it appears B:ht

if

the G a l s r k k ~ btsgratioa sf

%he

equations of amtion i o taken over only cne half wave in b t h

wira

and eircumfersx~tf& dirsetions, the

ot bs said to apply %g% a Breathing n3&e

vibrasion of

o

circdar cylbdrla~al @hell. P ~ Q W ~ V ~ P 1% the Gdericin %x&egration which Ped $0 eqmtion 2 . 2 6 Isad beein carried m s ~ = a &d.l wave in

b*&

and ciscamfereatid &rectfono, an spuatfo~x quits dufssent from 2.216 w u l d h v s been fassid. I.n 4ae$:

%he

quad~aatic nordhsarity v86ulbd havs been gone, leaving a forra of

M f i n g ' B equa%fon. &ad the objeetionabke m0z2ent dong the b s ~ u ~ d - aries Y = cosetmt w ~ o d d haye depa-ed vdtB quadratic yrp,onlfr~-

aaaitg

.

Bug, t3:a h$erestbg " 'bucuing" @ezo:mena and &e d @ f e ~ e ~ z c e of tiArne dmPa5 b\x~a~d defiaction from time daring outward deaectio~ will &843 be r3i 81ag from the a a l y sic

.

E

i

the a a b g ~ l s wae %o start vdth equation 4.2 %ha o a f

variation B s h g intzsducsd by linteg~athg over the? doamain

-

L

s x 5

t

a

-a

L

-

\I' L 52

,

ithe r e s d t i n g ~ q ~ ~ f $ e i o n ~ ~$331 $@odd be limited to

r) -4

o ~ d s r to r d i s v e this ~estsfctton use

M M

be w d e of

a

noalhear germ 0 4 the rscmtlg developed MorPsy e ~ a k i o n s , En his paper (ref. 9 ) :&arr$$3y &&o ~ h o w n that

a

slight rn~&Siea%fan of the Don-.

v d a e e of W in %%&tie problems. This eoncluoion ha@ been sub- stmtfated in a ppawr by Houghtoa

a

d

.

Jobas (10) whare Morl@%iyq s @qua$fon i a c ~ m p a ~ a d no* o d y d t h &rn81'@ equatioz bat al.1~6~ with the 6;?quathoao b a a r h g the n m o e Biieseao a& Qrammel, Vlassov, Tinla shenko

,

B i j l a ~ d , Haghdi

and

EBPPY and Kemard

bs applbsd. l.f

a

severs doubt abmg the h:ozl@y sqwtfsae f held by

%;he

r @ & d 6 ~ ~

he

zmay

s M a h

the BQA=~S @quation x @ @ u ~ % @ by drop-

@

piag ane t s ~ m in fbd e ~ a t i o n ,

T h e boaandory condition@ are

The rd~eming of the boundary csxaditiora8de on S are the ~ m 3 e 8 s in

section 2.2* The meaning sf $ha bow2dary eonditisals on

F

are s i m i i l o r ts that of section 2 . 2 , bat B e y s they are true on the aver

-

age over the entire boundary where, before th@y W@FB Om8 on a point by p i n t bagis throughout %Be b s u d a r y .

Fdow, uisea a dull cylinder is considered, N q l =

a E

and

and the associated stress fw2c$ion, which satisfies c~nlpatibiility,

t h e squili$siu~-zs squation fs~sra squatigsao 5.6 produces

-

L E h

rr

[%

b

+

*[DL(EP+

(w(-$)']

+

2

[ ~ ) ~ [ ( % ) ~ ( % ) 1 1

Equation 5. 3 is to be weighted with equation 5 , 2 a d the CaSHerhin a v a r a g b ~ g integral bo to be carried sue owes t h e domain -1 4

x

_{L ;}

*&ere all ather integrals is &his CaEerakiaa average v a l s h -

The mlbajrkin i n t e g r a t i ~ n of the equilibrium eqaatisn than produces

Once again the notation of equatioa 2 . 20 i~ used with the exception

*

that G is now replaced by

-

G z

3,

-

IYnen these e x p r e ~ ~ i ~ n s are used to simplify equation 5.5, equation 4.5 is used to i%~%roduee $be aspect ratio, t h e nondin1anoiom1 amp- litude is governed by the equation

where

ln a receat -=per f l ~ e f . 6 ) , Chu has obtabsd a ~ s i m f l a r equation by a different process. h e collects t e r m s mti;.hich contain first haz- monies of t h e space variables from an equation equivalent to equa- %ion 5.3 and neglects the higher harmonic apace? terxns.

By

this procedure he obtains ar equation which differs from equation 5 , 7

by a factor sf two in the cubic t e r n . Ws\v@ver, since the h t a n t of h i a paper was to shovi the t r e n d s of frequency dependence on tbs amplitude of vibration, the l o s e of this factor of two s h s d d not In- validate h i s general concluefsns

.

Of course it mast be noted tha$ G h u h development does not start from the 3-d;arlhey equation a d so has itlstead of

X r M

in his development of the equa- tion,

Because equagisn 5 . 7 i~s a form sf Ddfing' a equation, a great deal of i d ~ r m > a t j b ~ n a b u t i t @an be obtained from the Hieer- a t u r s of aonlineas m.stsiecBanfc s

.

Since this i d o rn2ation is gene r a l l y

"4h available, it is & 4 ~ 1 ~ ~ e $ sary $0 repaat the derivations here

.

It

vvil be the work of the next aweral sections to make use of the already h ~ w n data for M f i n g s s aquation in order to discuse the

Equation 5 . 3 will be written in the as;i:>~plified form

Sbgulorities of equation 5.8 %viBl be at the sso%s of

P

-

Bat since d3 O the only singularity i o a sCeqb1e center at

4

= O

.

*#here the subscript '¶ ddLts on the period syn3boB is to designate @2$rt

the &Lorley eoplaticn has been used. T t e limits and are the read roots of

-47

-

where the sign has been chasela so that

4

shld d a r e real. The

F 2 C i & b %Ire aid of equations 5 . 4 4 land 5.15, equation 5 . 4 2 may be evcP- uated foLlo\yuing the me&od presentee in Ap2endk

I.

5. B

w - _{a he resimwse}

0% the P d f i n g equation to h a r m i a ~ ~ ~ i c excitati~n

k a a received considerable attention. The %ollopabng procedure % e ts be fa-2 in ref. 41 and i o r e p a t e d bere with only sufficient detail t o nsaka the translotion ts the notation of the present probParn -der- atandable.

The frequency used t o n~ndi~~enssionahbae the tiire w % k L be chosen t o be

diM

so %hat = iJ,,

t

and Q: =,

I

agci in a&db

-

tion the fr@qu@~%ey in &be deacmiiaation s f d is; also L ~ L M

.

GBLI-

o i d e ~ the case ~w11sra the coefficient d is a q u a ~ t i t y ~rna1.l relative &a unity*

Let forcing f w ~ ~ t i o n be

wahere

1

has been added t o each side of equation 5.6.

8I.f the investigation is in the v i c b i t y of the natural f ~ e q u e n c y of the linear systenl, f

.

e .

,

if the bveetigatfon f s to seek the condf

-

tiorno neaP reoonmce, than

2

is close to unity a d ( ;\'

-

)

be small of order

d

.

?in tb& caae d1 quantitiee on t h e right haad aids of equation 5.47 are o m a U sf order 6 Under these condi-

tiona, equatisn 5.33 is an instournen9 for ~ M a a i n i i g an agpra>tin;.,a&a eolasahioa by iteration. Let the iterative solution ba

where

&

4 4

4

_First L~t?troduce

$=

4

into equation 3.17.

K - 1

Thew drop t e r m s of order d ~ 9 the result h

*

-4ccoadLng ts Sgcker it cam be s h o r n that only C O S

3

' M (where 1

-49

-

where 4. is a c o n s l a t . M r o d u c h g

(

=

+ $

into sqra~tion 5. 47 give8

The term@ in squara brackets In equation 5 . 2 2 are second order t e r m s and as such n3ay be neglectede Then subetitatbg equation 5 . 2 1 into equation 5 . 2 2 and expanding t k s cosine cubed term

% the coasficient of @ o S 'YM i~ nsn-zero then $ha o s l u t i s a to

equation 5.23 will co~rgtsirra, a secular t e r m which is of the form c

.

Since Ohis w o d d be a non-periodic aolutisn i% ia not acceptable md is prohibited by sa9tLqg

[ ( X % I ) A , + d q O

- 2

3

=

0 .

Z

.3olvbg for

A

-50-

&%

Aceorang to S g 0 k 8 ~ the choice of 8etting ,&

,

5 hAo $8 ""dg~isive"

at thi.8 point a d Qhe iteration may p r a ~ s e d 8tiwaya applying Ohis principle. E

A,

Z at this p i n t * then

Gusavea off

I

A 0

1

ao a gunctisn s f

A

,

$02 d = 0.04 are sH-oa,m,

in figure 14, Note that the central curve wlasked

9,

= 0 in big- upe 14 depicts the free vibration dependence ob

2

on A, as d e t s ~ m h e d by Bhs "D*dfbng9' "eration scheme. To test the rak338

of validity of equation 5 . 2 5 (bvith

go

= 0 ) it ehsuld be compared with tila result 8 sf eqaticra 5.12 us@ #Mbx =

.

Tihen the respsase c u r v e .ha@ multiple values of an3pBituda M ~ f c h casrespond to a single value of t h e driving frequency, or in the n & ~ t i o a used here, to a afngPe value ~f

2

,

the gossibfSllty of a jm2j% phanepa3ena may exist. The jump csuass the ampleude off viibratio~~ to chmge draa~atlcally without a cha9ge in dr%vi&g Pgequsncy. B* it t o not

an

instability in

$Be

sense of unbouwded m p l i t u d e ; L~deed, it reduces the amplitade of vibration.

T,

X.

t*

Caughey has chovm that a necessary a d a d f f c i a t conditiow for the existence sf t h e J m p pl1er~omer~a is that the ampli$ude fpequency reaponas cumes 2 o a ~ e ~ s a vertical tangent. Xn addition he haa

-51-

of verkicd tangent$, The loci of v e r t i e d t a g e n c y in the Limitk2g case of v m i s h b g l y small a m p i n g are determined by BHs zelationa

which b t l i corn@ ~ C O the value

'X

-.

4 for A, = 0 .

terion cf stability is the following: lot {(y) be a a o l a t i ~ n

of

a

solueion of the sanae equation; and let

5

+ l z ) be a h b i t r a ~ i l y small

a(some time

TO

It

S

i ( 2 ) coatinuew ta be arbitrarily small f o r all other values of '2'

.

then fie soPation

#

(2) i,o said be stable, oti~er~vLse it iis ~je3stabla.

Xf tlis criterion is applied tc equatiori 5. i? vr&h keo =

=

C],

cos

(/z y )

.

the equstiou governling

6

&

( Y 1 i s the

The harxncnic solution

4~2)

~ v a o obtained under the tsslrinp- tisn tBat

d was

szma11, thsrsfora this aaaumy%ion maat be carried

into squs$ion 5.29. h addition r n w t satisfy equation 5 . 2 5 .

Gs:mbiaatLone sg % amd A for which the aslution to equatiou

-92-

foJilauiifag t r m e f a r ~ ~ a t f o n is made

g =

ZA'CM )

then e ~ a t i a n 5.29 becomes

It

equation 5.25 is ueed wigh equation 5 . 3 5 to eliminate in t h e

-

expressions lor

$

and Uien

-53..

5 . 8

S h c s t h e subharmonic Psspcnae of the W f h g eqaatlsn i;s well docmmentsd for

a

e u b h a ~ r n o n i ~ of order 1/3, the details s E $be d e r i v a t i ~ n will n& be given here, Consider eqaation 3.8 wr&esn in the form

where Y = 43 & ana9

2 2

= dz/dz. B is desired t o find the ccn- dition mder v ~ k i c h a a.olattan of the form

can exist. U p o w htggsdacing equation 5 . 3 5 into equation 5.34 \;3d

=

02q.

These may 5e rewrittan in $be form

-54-

Z

Now iff is eliminated m d the iteration of t h e e equatioshile i s started with

%hie subharanonig: vibration deve10p1 t h ~ ~ a g h %he branching of the harmonic vibration when

The surprising ~ b s ~ r v a t f o n that 8x1 uItr&~i$~monig2 of o$de%i"

2 e m exist in s nonlinear a y atem of the D d d h g type has been veri- fied by

T.

X. Cough,-y. h order to study the condiitiona ~ q d a r

%ich such a soliltion migM exist in &s ~ e ~ g m n ~ e of a cylindrical

shell, consider the eqmtion

V ~ ~ B F B /Yr, = J L M ~ and

x ( @ )

i s rtlon~idered to be arnsaltl in csmps;9-

i s ~ n to unity. To find $be d$r&srmorrzie of Q Z ~ . @ F 2 in the s y ~ t e m

rspraosnged by equation 5. 4 O B it i s necessary $0 look for a wr?,o%ion

of the fa~n2

- 5 5 -

U

@quatian 5.41 is h t r ~ d u e e d ia~to csquakfon 5 . $0 =d higher Bar;.- monies neglected, the foiBlo%ving r e s d t is obtained

Fsr

a.

nontzi~al aolutisn, the fs11owfng conditions

are

obtahed from equation 5.42

I t 1 [ 1 + $ q 3 ~

+ q f z ] =

0,

-56-

U s e of equation 5.44 to eliminate q' brsm eqaation 5.63b give o

Now since 2 L L I ha8 already bssa saa-ad it masy be u ~ e d to obtaie an approxDmate aolugioa. n u s equation 5.45 bscomea

S o db

61?

L L I

,

then

t e r m s of $he dsf~rmabion r s s p n e e gsoblsa?i ag a shell, eqaatiog~ 5.50 h&cateo that when a circular c y l b d r i c d ehall i o subjected to a fore b g fmetbon a& ona-ha16 the ata as all frequency 02 the squivo- %ant linear system, it i a p s ~ ~ i b l e to g e t a ae3ponss %-at includes o pup@ dilatation as wslB a e %ha P-damant&% and a I a t g a rsopomas

ag

The sqaatioa~ of motion are not d e t s s m h e d kg the nw;"absr sf madea u1ed. Therefor@, eguatiwas z ~ o t i o n are ehe same here as usad kg Caai3te~ I%& lt thae its, $he M 0 ~ 1 e y %~qaa%ians.

6.2

Tae b w d a v y condigioas sf @ecbioa~ 5 . 2 are applied awpa- ralelg to each

d

U e tcro m&e functions

S

(nt, n ) a8d @, _ZJ

m d to the @ s i m p s i t e stzess fwetion F which 6s essochted wit'&

The raodaa to bs used are

-

A

B

4,

c n l - ~ ) r (++-) H~

+

n

B

F,

- m u

-

R B ~ , (*td9~ 1 (-

.,]

=

~ 3 :

L a L a 36 L a

Xn Ohis the gumtities /-I i are

*

2 4 = c 0 s

(TX)

s I*, ( f l . ? ) c o s

(2))

Silbsn equation 6 . 2 is weighted with Sr*l

(qkO,(g~

and ineegrated

Now if the ao$.a$Hone

are used &s squatton obtained L F O ~ equation &. 6 is

The aacond equation may $8 obgiibined by I ~ e r ~ h a g e ~f the mode

n u m b s a v a ~ f a b ~ s s ( m , n ) i ( a ) a d v i c s v s r s a . U $ & f s

is dons and a ,more ~izmplified no%a$%on i n used then the &WO mode

and

f

i s ~ M a i n c d faom aquation 6.6a

by

replacing V by n ,

d by WI

,

and vice v e r ~ a .

43.6

E

the natural fsequene tea &H ( M I II ) m d

dLLA

(A,

V )

are well separated the eouplbg of @quaB%ana 6 , 9 i o very V J B ~ C ax- 1893

Y{

> >

,d+ and

$

>7

,&$

.

% tPd@s@ eoe@I~ienta are oh the same order td~mgwitude and if &She sxcitaeioas ape periodic with. a fregaagcy near

dLM

(m,

.

POB i n g t a n ~ e , then equations

2

s b c e

#

>

Z

5

under these ccnditioes (see &gar@ i b ) . The main respoapre is in the

4

mode s v h i ~ h 18 oupled from the infiuencgs

of

the

&

made. At the a m @ 0 6 ~ x 1 ~ eke respoass

d

the

6

hnode is excited in part by the

d

mode, but its aanpli- t d s i~ still a m d l c o m p s ~ e d t o its ;iecompaaion mode.

ltf

the &pace df atribate& ag the ~ O T C in8 Pmctfon P(rJr'; t ) is such that

QG

>>

Q_Q than equations 6 . 1 4 fuahssr oimplffy t o

Bat in this instance the

@

equation admi% s soiliaSon

-69-

Araold and TiorbuHon (43) lio presented gar one thickness r&io (h/ak = .004. Thia plot show9 q d t e dafinigely Ohat f o p a = Z 2 ,

m

rra

A nd

(

)

=

[

)

=

2 zuad

t

'

=

20, the two modal Irequencies

4 7ra

ere

-

not separated. But POP n = 2,

(%+)

=

(-

)

=

5*

d

=

20

that gar aa uac ouk~led system.

The scrppling may alsc be weak if

y+

d

f+

aae small compared tc

@+

and 6'4 r s s p c t i v e l y . Eavierer, an examina- tion

of

&eee

four

m r w & e r @

i a d t ~ a t e a %hat in general such

a

rehtioaniahh3 I s hard $0 aehisvs.

in which

&

Cc

/

and

,

A

,

r

A 4

Ad

are not large c o n -

1

-

pared to u ~ i t y .

Tbs atody 0% &s e g o m o d e s a y s t s ~ ~ will S018ow the work 0%

T, K. Gs~ghsy (42) and So ~ X ~ B H T ~ S ~ V ~ ~ @ p & i t i ~ n of his d @ ~ e l ~ p -

meno %xi11 be iac1uded. The genera4 m~b50d IB e ~ e n o f o n ao% the? M q P o f g a d B ~ g M a b f f method to the forced respoase of a two mods modinear system. h ~ o l u t f a n t o B~ULO~BBI 6. % 3 1s f b ~ ~ k e d for in the for,-

as a conaequsace of Qhs sequisemant that &a aoPa%los be periodic* &a addition A 4 )

44

,

<&

d

f

4

are viawed a a slowly vary- ing functions cf Y

.

Because of this. & & , f l d , < d , & d and. their first desivaei-ases mslr b~ replaced

by

their average value over one cycle in coonputhxg the responoa of the syotsra, Taa average vala@8

are

deadad by t h s eupsrscaipt bars. Thus t h s ~ a average8

The oteady state condition is

- 7 4 -

dBf%arez%t or nsn-synchronous sources, The p a ~ a . ~ e t a r s

d

<